Abstract
A. H. Stroud has shown that n + 1 n + 1 is the minimum possible number of nodes in an integration formula of degree three for any region in E n {E_n} . In this paper, in answer to the question of the attainability of this minimal number, we exhibit for each n n a region that possesses a third degree formula with n + 1 n + 1 nodes. This is accomplished by first deriving an ( n + 2 ) (n + 2) -point formula of degree three for an arbitrary region that is invariant under the group of affine transformations that leave an n n -simplex fixed. The formula is then applied to a one-parameter family of such regions, and a value of the parameter is determined for which the weight at the centroid vanishes.
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